Optimal. Leaf size=71 \[ -\frac {x e^{b^2 x^2} \text {erfi}(b x)}{\sqrt {\pi } b}+\frac {\text {erfi}(b x)^2}{4 b^2}+\frac {e^{2 b^2 x^2}}{2 \pi b^2}+\frac {1}{2} x^2 \text {erfi}(b x)^2 \]
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Rubi [A] time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6366, 6387, 6375, 30, 2209} \[ -\frac {x e^{b^2 x^2} \text {Erfi}(b x)}{\sqrt {\pi } b}+\frac {\text {Erfi}(b x)^2}{4 b^2}+\frac {e^{2 b^2 x^2}}{2 \pi b^2}+\frac {1}{2} x^2 \text {Erfi}(b x)^2 \]
Antiderivative was successfully verified.
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Rule 30
Rule 2209
Rule 6366
Rule 6375
Rule 6387
Rubi steps
\begin {align*} \int x \text {erfi}(b x)^2 \, dx &=\frac {1}{2} x^2 \text {erfi}(b x)^2-\frac {(2 b) \int e^{b^2 x^2} x^2 \text {erfi}(b x) \, dx}{\sqrt {\pi }}\\ &=-\frac {e^{b^2 x^2} x \text {erfi}(b x)}{b \sqrt {\pi }}+\frac {1}{2} x^2 \text {erfi}(b x)^2+\frac {2 \int e^{2 b^2 x^2} x \, dx}{\pi }+\frac {\int e^{b^2 x^2} \text {erfi}(b x) \, dx}{b \sqrt {\pi }}\\ &=\frac {e^{2 b^2 x^2}}{2 b^2 \pi }-\frac {e^{b^2 x^2} x \text {erfi}(b x)}{b \sqrt {\pi }}+\frac {1}{2} x^2 \text {erfi}(b x)^2+\frac {\operatorname {Subst}(\int x \, dx,x,\text {erfi}(b x))}{2 b^2}\\ &=\frac {e^{2 b^2 x^2}}{2 b^2 \pi }-\frac {e^{b^2 x^2} x \text {erfi}(b x)}{b \sqrt {\pi }}+\frac {\text {erfi}(b x)^2}{4 b^2}+\frac {1}{2} x^2 \text {erfi}(b x)^2\\ \end {align*}
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Mathematica [A] time = 0.02, size = 63, normalized size = 0.89 \[ \frac {\left (2 \pi b^2 x^2+\pi \right ) \text {erfi}(b x)^2-4 \sqrt {\pi } b x e^{b^2 x^2} \text {erfi}(b x)+2 e^{2 b^2 x^2}}{4 \pi b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 58, normalized size = 0.82 \[ -\frac {4 \, \sqrt {\pi } b x \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} - {\left (\pi + 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )^{2} - 2 \, e^{\left (2 \, b^{2} x^{2}\right )}}{4 \, \pi b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {erfi}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int x \erfi \left (b x \right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {erfi}\left (b x\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 66, normalized size = 0.93 \[ \frac {\frac {b^2\,x^2\,{\mathrm {erfi}\left (b\,x\right )}^2}{2}+\frac {{\mathrm {erfi}\left (b\,x\right )}^2}{4}}{b^2}+\frac {\frac {{\mathrm {e}}^{2\,b^2\,x^2}}{2}-b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{b^2\,\pi } \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 63, normalized size = 0.89 \[ \begin {cases} \frac {x^{2} \operatorname {erfi}^{2}{\left (b x \right )}}{2} - \frac {x e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{\sqrt {\pi } b} + \frac {e^{2 b^{2} x^{2}}}{2 \pi b^{2}} + \frac {\operatorname {erfi}^{2}{\left (b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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